Why do we need convexity adjustment (CMS, ...)?

[Emanem]

Hi all.

My question is, why do we need convexity adjustment when we need to determine the FWR for a CMS shaped leg?
I've read both Hull and Brigo/Mercurio; the mathematical explanation is ok, but I really don't understand why. The how is ok (and I don't even care that much - ie not a mathematician).

Now, assume we have our beloved risk free rate zero curve, from now to 50yrs.
Let's take two specific swaplets of a CMS leg that pays every 6 months (at end of accrual period) the 5yrs SWAP rate and this leg will last 1yrs from today.
T0 = today
T1 = today + 6 months
T2 = today + 1 yr
We'll have two cashflows:
1) in 6 months time (T1) we'll recv T0 5yrs swap rate on 6 months accrual period
2) in 1 yrs time (T2) we'll recv T1 5 yrs swap rate on 6 months accrual period
Now, my understanding is the following:
1) As seen as T0 5 yrs swap rate is out there on the market we don't apply any adjustment, or if any, this would be 0
2) In this case we should apply a convexity adjustment, because we can't just take the 6 months FWD 5yrs swap rate. Why this?

I've managed to read between the lines that (2) perhaps is due to the fact that the 5yrs swap rate would accrue each 1 yr, while in our case the accrual period is shorter (6 months). Probably I'm wrong, but if this is one of the reasons, again, why?

If the above is the only case when this happens, what would happen if our CMS leg would have instead of 6 months accrual period, 1 year as in the underlying 5 yrs swap? Should we still apply some convexity? Again, why?

Probably, no definitely, I'm wrong. That's why I'd like to have some insight!
Thanks again,

Cheers,

 

[Emanem]

Hi Maciek,

thanks for your answer; I'll look at the paper as soon as I have some time. Still related to my example, could you please motivate why in the two CMS example legs we would need conv adj?

Thanks again,
Cheers!

 

[Emanem]

Hi, after reading the paper (well done btw) I understand even the why.
So probably I should formulate this thread/question in another terms: apparently my knowledge of zero curve is wrong! :-P

When we do build a yield curve (zero curve), we should be able to:
1) Get value of at par swaps starting today (or today+2)
2) Get discount values for future payments
3) Get forwards rates
4) Get fwd swaps (ie.5y swap rate starting in x time)

Am I correct? In case I am with (4), we should be able to (easily) get the at par value of the (as example) 5 yrs swap starting today, or in 3 months, or in 6 months, or 9 or 1 year without any adjustment.
Where am I wrong?
This is why I am asking about CMS adjustment. Because if my above sentence is correct, why do we need to adjust these values?

So, as seen as conv adj is needed, where my knowledge of yield/zero curve is lacking?

Thanks again,
Cheers,

 

[Emanem]

Hi Emanem,

Let me explain this without writing any formulas and going into the bootstrapping methods. I hope this will help:

1) You first get the spot swap curve (you sum discounted cashflows and adjust the rates so that the swap price =0. In fact you just take the swap rates from the market- brokers, bloomberg as these rates are quoted).

2) So now you have the swap curve which is your zero curve. You want forward curve. let's say you want to calculate a forward rate from t2=2year to t3=3year. So you want a 1year forward starting in 2 years . Assume your zero curve is normal (increasing rates with maturity). So you know that the 2year spot rate is bigger than the 1year spot rate.

3) Now look at the relation between bond prices and bond yields in the graph attached (it's the same kind of relation as between swap rate and swap price). Keep in mind that according to your current information in 2 years from now your 1yr forward will be your spot rate. This is not however fully correct as the whole spot curve would have to be recalculated again (i.e find rate that makes discounted cashflows=0) as the swaps will not be at par any more! This is because the froward curve (effectively the then spot rate in 2 yrs from now) will be different from the current spot rate. So if you calculate swap price starting in 2yrs from now using a spot rate obtained from your forward rate it will not be equal to 0. The swaps will not be at par! Hull's convexity adjustment is adjusting for that.

I hope the above helps to get a big picture but it is difficult to explain this in words. I will not write formulas as they are all in Hull and in the paper I quoted.

Maciek

THANKS
I get it finally... after some years I was wrongly assuming that with zero curve we could easily get the fwd rate without worrying about anything.
I was basically missing the second part of point (3).
But, as you said, as seen as the 1 year 2yrs FWD rate should be adjusted because (in normal conditions) the 1 year rate (0 years FWD) is (generally) lower.
Hence we adjust it of some factor depending on the volatility we expect for the rate to be for that period. Am I correct with my logic?
Because now it does make sense to have an adjustment.
And this is why this sort of thinking is valid for all instruments which are CMS like...

Let me know if my logic is wrong (hope not).

Thanks again!

 

[Nandita]

Hi

I'm doing a presentation at work on Duration of Bonds & Convexity .

One question that puzzles me is the fact that why can't duration in reality be linear?Why would they end up being convex?

I don't come from a Quants background, so an explanation in simple language will be very helpful.

Many Thanks

Nandita Rao
Sydney

 

[Sanket Patel]

The relationship between price and yield is not linear. That said, I think the wording in your question "why can't duration in reality be linear" is a bit off. Duration is a first order measure; that is, it measures the 'linear change' of an otherwise nonlinear linear function.

The price of a bond doesn't fall into a simple linear y = mx + b type function. If that was the case, you would see a perfectly linear relationship - for every 1 unit change in x (the yield), you would observe an m-unit change in y (bond price).

That, really, is about as simple as it gets.

 

[Emanem]

I think you got it right now

If the curve was flat (theoreticaly it can happen) then we don't need this convexity adjustment.

As it s stated in the paper I quoted in my previous post- this is just one typy of convexity adjustment (the most importat as the other adjustments tend to cancel each other out). In some big banks there are separate departements where all what they do is calibrating spot and forward curves.

Hi, one last question then.
Even in case of a simple FRA, in order to determine the FWD rate (i.e. in a FRA that starts in 2 months and latsts for 1 month), we should apply the same kind of Convexity Adjustment.
Am I right?
Then we don't do it because maybe is so negligible, but in theory we should.
Am I right?

Cheers!

 

[Tumisho Mabelane]

I'm also trying to understand the concept of convexity adjustments. Will somebody please give the name of the "quoted" paper for further insight?

 

[A modest quant]

I'm also trying to understand the concept of convexity adjustments. Will somebody please give the name of the "quoted" paper for further insight?

I am also trying to understand some of the market phenomena (convexity adjustements, implied vol skews, CVA, etc...) which many textbooks donot explain clearly and consistently. These are indeed phenomena because they are not consistently explained by the current risk-neutral approach. A good exposition can be found in "Intrinsic Prices of Risk" by Truc Le here: at http://ssrn.com/abstract=2406501

 

[Ole Bueker]

The way you post about the paper, together with the fact that your name is "A modest quant" (the paper you cite writes "This paper is a modest attempt to prove" in the abstract) makes it seem that you are the author of the paper.

If that is the case, I would suggest you at least cite yourself. (Not that I am against you advertising your paper, just state it clearly)

 

[A modest quant]

Hey Ole, it just happens that I'm indeed a modest quant and a humble trader and successful one...mind you.
I'm also indeed in favor of the paper, however I'm not the author. Perhaps, I suggest a student like yourself should spend more time in the real world than read lecture notes offered by your lecturer. Am I right?

 

[financeguy]

Just came across this thread due to the paper marketing and realized the convexity adjustment question hadn't been answered in layman's terms. It's a weird almost circular concept that exists only in rates markets. If you were to price a receiver swaption and a payer swaption struck at the forward rate, you would think they should be worth the same amount. They are not - because in the case you end up in the money on the receiver, your future cash flow gets discounted at a lower rate (obviously, because rates have to be lower for you to be in the money), and vice versa for the payer. The receiver is therefore worth more than the payer. Taking this outside of the swaption space and considering only linear swaps, if you were just as likely to end up below the forward as above, you should choose to always choose to receive the market, because on expectation the strategy has positive expected value. Imagine a normal distribution of outcomes centered around the forward, but on the left hand side the payout is slightly greater because of the lower rate of discounting. Therefore, the forward rate isn't actually the market's exact expectation of a given rate in the future. The rate has to be adjusted down for this discounting effect, and the way to do it is to find the forward rate at which receivers and payers price the same, which of course depends on the implied vol of the swaptions. This adjustment from the forward rate to the rate at which put-call parity holds for swaptions is what the market calls the convexity adjustment to find the true value of expected rates in the future. I call it a weird and circular concept because it feels unnatural that options prices should tell us something we didn't already know from the underlying itself, but there it is.

 

[A modest quant]

Just came across this thread due to the paper marketing and realized the convexity adjustment question hadn't been answered in layman's terms. It's a weird almost circular concept that exists only in rates markets. If you were to price a receiver swaption and a payer swaption struck at the forward rate, you would think they should be worth the same amount. They are not - because in the case you end up in the money on the receiver, your future cash flow gets discounted at a lower rate (obviously, because rates have to be lower for you to be in the money), and vice versa for the payer. The receiver is therefore worth more than the payer. Taking this outside of the swaption space and considering only linear swaps, if you were just as likely to end up below the forward as above, you should choose to always choose to receive the market, because on expectation the strategy has positive expected value. Imagine a normal distribution of outcomes centered around the forward, but on the left hand side the payout is slightly greater because of the lower rate of discounting. Therefore, the forward rate isn't actually the market's exact expectation of a given rate in the future. The rate has to be adjusted down for this discounting effect, and the way to do it is to find the forward rate at which receivers and payers price the same, which of course depends on the implied vol of the swaptions. This adjustment from the forward rate to the rate at which put-call parity holds for swaptions is what the market calls the convexity adjustment to find the true value of expected rates in the future. I call it a weird and circular concept because it feels unnatural that options prices should tell us something we didn't already know from the underlying itself, but there it is.

I don't quite agree with what you said. The fact is that the term "adjustment" indicates that what we do in practice doesnot follow theory, therefore we realized the discrepancies between contracts on the same underlying. This doesnot only exist in rates world, but also in equities world. The same theme repeats with CVA, DVA, etc... at times i myself have made money off these arbitrage opportunities...

 

[financeguy]

I don't know what you are talking about

 

[A modest quant]

I don't know what you are talking about

Of course you don't! If everything going on in the financial market can be explained and understood by your layman's terms, then financial mathematics departments never exist. Your explanation on convexity adjustment seems to be a resounding words of what you have heard from someone else who is likely an academic.

 

[financeguy]

In this imaginary world in which you are the trader and I am the academic quant and not the other way around, how are you going about arbing the market?

 

[A modest quant]

In this imaginary world in which you are the trader and I am the academic quant and not the other way around, how are you going about arbing the market?

A simple realistic arbitrage opportunity lies between a forward contract and a futures contract. While both supposedly have the same payout at maturity, their real prices are almost always different: one being priced in forward market, the other in futures exchange. Certainly this is an arb opp, but so often that the cost involved in exploiting it outweighs the profit, therefore their prices remain different. If you still don't understand this simple realistic scenario, i suggest you work with a rates trader who trades futures, he will tell you all about this. Goodluck!

 

[financeguy]

I'm familiar enough with FRAs vs futures. But that's just because when you trade futures you settle up daily pnl on the exchange, while when you trade a FRA you only settle at expiration. Whether it's worth it or not to be received of a FRA and short the future, or vice versa, is a view on volatility. FRAs get discounted at the prevailing rate at expiration, so just like what I posted above, if you are received of a FRA and rates go down, you are happy that rates have gone down and your winnings are discounted at lower rates, and if you are paid FRAs you are happier than you'd have otherwise been as the money you've lost is discounted at a higher rate. Exactly how much happier or sadder is determined by how far rates have gone up or down. Futures on the other hand are perfectly linear. This is definitely not an arbitrage opportunity, and the convexity adjustment that exists between FRAs and futures is just a function of the implied vol of swaptions, costs of funding for each counterparty, and the amount charged for collateral on the exchange. You can make or lose money going either way on futures vs. FRAs, but it's not an arb at all. The prices are different for very good reason, convexity adjustments to go between futures and FRAs are quoted all day long and are well known among dealers, and they don't necessarily present an amazing market opportunity.

 

[A modest quant]

I'm familiar enough with FRAs vs futures. But that's just because when you trade futures you settle up daily pnl on the exchange, while when you trade a FRA you only settle at expiration. Whether it's worth it or not to be received of a FRA and short the future, or vice versa, is a view on volatility. FRAs get discounted at the prevailing rate at expiration, so just like what I posted above, if you are received of a FRA and rates go down, you are happy that rates have gone down and your winnings are discounted at lower rates, and if you are paid FRAs you are happier than you'd have otherwise been as the money you've lost is discounted at a higher rate. Exactly how much happier or sadder is determined by how far rates have gone up or down. Futures on the other hand are perfectly linear. This is definitely not an arbitrage opportunity, and the convexity adjustment that exists between FRAs and futures is just a function of the implied vol of swaptions, costs of funding for each counterparty, and the amount charged for collateral on the exchange. You can make or lose money going either way on futures vs. FRAs, but it's not an arb at all. The prices are different for very good reason, convexity adjustments to go between futures and FRAs are quoted all day long and are well known among dealers, and they don't necessarily present an amazing market opportunity.

Both you and i observed the price differences in FRA and futures, but interpret it differently. Lets leave it as that!

In another post "Should a sophisticated model..." you agreed with CasanovaJ who mindlessly said "What does vol have to do with futures? ". Here you said "... short the future, or vice versa, is a view on volatility." You're full of contradition, financeguy! ...and "...the convexity adjustment that exists between FRAs and futures is just a function of the implied vol of swaptions,..." What is this? You say this just because Hull White or some mindless professor said so?

 

[StevenChen]

Back to title, as I read Hull's book, the answer is simple,
function y=1/x is nonlinear, in this case y is bond price B and x is[(1+r)][/T]
Pricing a derivative is to deal with underlying price probability distribution.

Denote probability distribution of bond price as G(B),
and convert G(B) to find probability distribution of r as H(r),
we know G(B)!=H(r) due to nonlinear function y=1/x.
But according to theory, different derivatives with same underlying should have same probability distribution assumption. A convexity adjustment is made to deal with this disparity.